The first issue is that, due to the finite size of the sample, as
the value of \(k\)  increases the sum over the realizations is rapidly
dominated by the largest value in the sample. This leads to the artificial
linearization of the tails of the estimate of the scaled cumulant
generating function. Given \(\tau _{b}\)  and \(N_{b}\) , one can estimate
upper and lower bounds \(k_{c}^{-}(\tau _{b},N_{b})\)  and \(k_{c}^{+}(\tau _{b},N_{b})\) 
such that the estimate of generating function converges for \(k_{c}^{-}(\tau _{b},N_{b})<k<k_{c}^{+}(\tau _{b},N_{b})\) .
There are different possible ways to determine values for \(k_{c}^{-}(\tau _{b},N_{b})\) 
and \(k_{c}^{+}(\tau _{b},N_{b})\)  given a sample of data, and to estimate
their scaling with \(N_{b}\) , as discussed in [1]}.
In this paper we have taken a completely empirical approach, and determined
the bounds by requiring that the relative contribution of the largest
value in the sample to the estimate of the generating function does
not overcome an arbitrary threshold of 50% (see Appendix ).
Note that if the observable is bounded, that is if it has an upper
or lower limit, the linearization is not an artifact but it is the
correct behavior of the scaled cumulant generating function.
